10. The axis of a parabola is along the line $\mathrm { y } = \mathrm { x }$ and the distance of its vertex from origin is $\sqrt { 2 }$ and that from its focus is $2 \sqrt { 2 }$. If vertex and focus both lie in the first quadrant, then the equation of the parabola is\\
(A) $( x + y ) ^ { 2 } = ( x - y - 2 )$\\
(B) $( x - y ) ^ { 2 } = ( x + y - 2 )$\\
(C) $( x - y ) ^ { 2 } = 4 ( x + y - 2 )$\\
(D) $( x - y ) ^ { 2 } = 8 ( x + y - 2 )$

\section*{Sol. (D)}
Equation of directrix is $x + y = 0$.\\
Hence equation of the parabola is

$$\frac { x + y } { \sqrt { 2 } } = \sqrt { ( x - 2 ) ^ { 2 } + ( y - 2 ) ^ { 2 } }$$

Hence equation of parabola is

$$( x - y ) ^ { 2 } = 8 ( x + y - 2 )$$

\begin{enumerate}
  \setcounter{enumi}{10}
  \item A plane passes through $( 1 , - 2,1 )$ and is perpendicular to two planes $2 x - 2 y + z = 0$ and $x - y + 2 z = 4$. The distance of the plane from the point $( 1,2,2 )$ is\\
(A) 0\\
(B) 1\\
(C) $\sqrt { 2 }$\\
(D) $2 \sqrt { 2 }$
\end{enumerate}

Sol. (D)\\
The plane is $\mathrm { a } ( \mathrm { x } - 1 ) + \mathrm { b } ( \mathrm { y } + 2 ) + \mathrm { c } ( \mathrm { z } - 1 ) = 0$\\
where $2 \mathrm { a } - 2 \mathrm {~b} + \mathrm { c } = 0$ and $\mathrm { a } - \mathrm { b } + 2 \mathrm { c } = 0$\\
$\Rightarrow \frac { \mathrm { a } } { 1 } = \frac { \mathrm { b } } { 1 } = \frac { \mathrm { c } } { 0 }$\\
So, the equation of plane is $x + y + 1 = 0$\\
∴ $\quad$ Distance of the plane from the point $( 1,2,2 ) = \frac { 1 + 2 + 1 } { \sqrt { 1 ^ { 2 } + 1 ^ { 2 } } } = 2 \sqrt { 2 }$.\\